Causal Inference with Spatio-temporal Data: Estimating the Effects of Airstrikes on Insurgent Violence in Iraq

Many causal processes have spatial and temporal dimensions. Yet the classic causal inference framework is not directly applicable when the treatment and outcome variables are generated by spatio-temporal processes with an infinite number of possible event locations. We extend the potential outcomes framework to these settings by formulating the treatment point process as a stochastic intervention. Our causal estimands include the expected number of outcome events in a specified area under a particular stochastic treatment assignment strategy. We develop methodology that allows for arbitrary patterns of spatial spillover and temporal carryover effects. Using martingale theory, we show that the proposed estimator is consistent and asymptotically normal as the number of time periods increases, even when the propensity score is estimated. We propose a sensitivity analysis for the possible existence of unmeasured confounders, and extend it to the H\'ajek estimator. Simulation studies are conducted to examine the estimators' finite sample performance. Finally, we use the proposed methods to estimate the effects of American airstrikes on insurgent violence in Iraq from February 2007 to July 2008. We find that increasing the average number of daily airstrikes for up to one month results in more insurgent attacks across Iraq and within Baghdad. We also find evidence that airstrikes can displace attacks from Baghdad to new locations up to 400 kilometers away.